Figure 1: The distribution of element durations and inter-element intervals from the whale vocal sequences included in this analysis. The times are z-scored within each study to enable direct comparison.
Group | Species | Effect | 2.5% | 97.5% |
|---|---|---|---|---|
Mysticete | Blue Whale | -0.255 | -0.331 | -0.178 |
Bowhead Whale | -0.184 | -0.318 | -0.051 | |
Humpback Whale | -0.678 | -0.692 | -0.665 | |
Minke Whale | -0.278 | -0.294 | -0.261 | |
Right Whale | 0.309 | 0.278 | 0.34 | |
Sei Whale | -0.213 | -0.363 | -0.063 | |
Odontocete | Bottlenose Dolphin | -0.242 | -0.347 | -0.138 |
Commerson's Dolphin | 0.221 | 0.087 | 0.356 | |
Heaviside's Dolphin | -0.119 | -0.32 | 0.083 | |
Hector's Dolphin | -0.008 | -0.274 | 0.258 | |
Killer Whale | 0.121 | 0.003 | 0.239 | |
Narrow-Ridged Finless Porpoise | -0.304 | -0.338 | -0.27 | |
Peale's Dolphin | -0.333 | -0.489 | -0.177 | |
Risso's Dolphin | -0.42 | -0.448 | -0.392 | |
Sperm Whale | -0.234 | -0.241 | -0.226 |
Length | Position | ||||||
|---|---|---|---|---|---|---|---|
Group | Species | Effect | 2.5% | 97.5% | Effect | 2.5% | 97.5% |
Mysticete | Blue Whale | -0.255 | -0.331 | -0.178 | -0.064 | -0.087 | -0.041 |
Bowhead Whale | -0.179 | -0.313 | -0.046 | -0.751 | -0.789 | -0.713 | |
Humpback Whale | -0.678 | -0.691 | -0.665 | -0.193 | -0.201 | -0.186 | |
Minke Whale | -0.278 | -0.294 | -0.261 | -0.017 | -0.023 | -0.011 | |
Right Whale | 0.309 | 0.278 | 0.34 | 0.107 | 0.096 | 0.119 | |
Sei Whale | -0.213 | -0.364 | -0.062 | -0.105 | -0.139 | -0.071 | |
Odontocete | Bottlenose Dolphin | -0.242 | -0.346 | -0.138 | -0.084 | -0.111 | -0.056 |
Commerson's Dolphin | 0.221 | 0.087 | 0.356 | -0.106 | -0.118 | -0.095 | |
Heaviside's Dolphin | -0.119 | -0.32 | 0.083 | 0.019 | 0.01 | 0.027 | |
Hector's Dolphin | -0.008 | -0.274 | 0.258 | -0.001 | -0.01 | 0.008 | |
Killer Whale | 0.121 | 0.021 | 0.221 | 0.528 | 0.428 | 0.628 | |
Narrow-Ridged Finless Porpoise | -0.305 | -0.339 | -0.271 | 0.168 | 0.151 | 0.185 | |
Peale's Dolphin | -0.333 | -0.489 | -0.177 | -0.013 | -0.017 | -0.009 | |
Risso's Dolphin | -0.42 | -0.448 | -0.392 | -0.196 | -0.2 | -0.192 | |
Sperm Whale | -0.234 | -0.241 | -0.226 | 0.028 | 0.026 | 0.031 | |
Language | Effect | 2.5% | 97.5% |
|---|---|---|---|
Anal | -0.104 | -0.113 | -0.095 |
Arapaho | 0.03 | 0.02 | 0.04 |
Asimjeeg Datooga | -0.063 | -0.073 | -0.053 |
Baïnounk Gubëeher | -0.102 | -0.11 | -0.093 |
Beja | -0.066 | -0.073 | -0.059 |
Bora | -0.127 | -0.138 | -0.116 |
Cabécar | -0.11 | -0.12 | -0.1 |
Cashinahua | -0.1 | -0.108 | -0.091 |
Daakie | -0.131 | -0.141 | -0.121 |
Dalabon | -0.079 | -0.091 | -0.066 |
Dolgan | -0.13 | -0.139 | -0.121 |
English (Southern England) | -0.053 | -0.066 | -0.04 |
Evenki | -0.101 | -0.109 | -0.092 |
Fanbyak | -0.091 | -0.101 | -0.08 |
French (Swiss) | -0.05 | -0.063 | -0.037 |
Goemai | -0.124 | -0.138 | -0.11 |
Gorwaa | -0.125 | -0.136 | -0.114 |
Hoocąk | -0.099 | -0.109 | -0.09 |
Jahai | -0.062 | -0.073 | -0.05 |
Jejuan | -0.093 | -0.104 | -0.083 |
Kakabe | -0.123 | -0.135 | -0.111 |
Kamas | -0.096 | -0.111 | -0.082 |
Komnzo | -0.081 | -0.091 | -0.071 |
Light Warlpiri | -0.144 | -0.154 | -0.133 |
Lower Sorbian | -0.078 | -0.089 | -0.067 |
Mojeño Trinitario | -0.147 | -0.156 | -0.137 |
Movima | -0.014 | -0.023 | -0.006 |
Nafsan (South Efate) | -0.072 | -0.082 | -0.062 |
Nisvai | -0.064 | -0.074 | -0.053 |
Nllng | -0.098 | -0.111 | -0.086 |
Northern Alta | -0.069 | -0.079 | -0.059 |
Northern Kurdish (Kurmanji) | -0.022 | -0.033 | -0.011 |
Pnar | -0.021 | -0.035 | -0.007 |
Resígaro | -0.079 | -0.089 | -0.069 |
Ruuli | -0.039 | -0.048 | -0.03 |
Sadu | -0.124 | -0.136 | -0.112 |
Sanzhi Dargwa | -0.135 | -0.148 | -0.122 |
Savosavo | -0.052 | -0.062 | -0.042 |
Sümi | -0.09 | -0.101 | -0.079 |
Svan | -0.066 | -0.074 | -0.057 |
Tabaq (Karko) | -0.213 | -0.222 | -0.203 |
Tabasaran | -0.138 | -0.151 | -0.125 |
Teop | -0.17 | -0.181 | -0.159 |
Texistepec Popoluca | -0.07 | -0.081 | -0.059 |
Urum | -0.126 | -0.134 | -0.118 |
Vera'a | -0.152 | -0.163 | -0.141 |
Warlpiri | -0.147 | -0.157 | -0.137 |
Yali (Apahapsili) | -0.191 | -0.202 | -0.179 |
Yongning Na | -0.128 | -0.142 | -0.114 |
Yucatec Maya | -0.067 | -0.079 | -0.056 |
Yurakaré | -0.198 | -0.204 | -0.191 |
Length | Position | |||||
|---|---|---|---|---|---|---|
Language | Effect | 2.5% | 97.5% | Effect | 2.5% | 97.5% |
Anal | -0.105 | -0.114 | -0.096 | 0.1 | 0.091 | 0.109 |
Arapaho | 0.03 | 0.02 | 0.039 | 0.099 | 0.09 | 0.109 |
Asimjeeg Datooga | -0.063 | -0.074 | -0.053 | 0.186 | 0.177 | 0.196 |
Baïnounk Gubëeher | -0.102 | -0.11 | -0.093 | 0.096 | 0.088 | 0.104 |
Beja | -0.066 | -0.073 | -0.059 | 0.065 | 0.058 | 0.072 |
Bora | -0.127 | -0.138 | -0.116 | -0.131 | -0.14 | -0.123 |
Cabécar | -0.11 | -0.12 | -0.1 | 0.021 | 0.012 | 0.031 |
Cashinahua | -0.1 | -0.108 | -0.091 | -0.019 | -0.028 | -0.01 |
Daakie | -0.131 | -0.141 | -0.122 | 0.164 | 0.155 | 0.173 |
Dalabon | -0.079 | -0.091 | -0.066 | 0.165 | 0.153 | 0.177 |
Dolgan | -0.13 | -0.139 | -0.121 | 0.043 | 0.034 | 0.052 |
English (Southern England) | -0.053 | -0.066 | -0.04 | 0.05 | 0.038 | 0.062 |
Evenki | -0.101 | -0.109 | -0.092 | 0.042 | 0.033 | 0.05 |
Fanbyak | -0.091 | -0.101 | -0.081 | 0.161 | 0.151 | 0.171 |
French (Swiss) | -0.05 | -0.063 | -0.037 | 0.16 | 0.151 | 0.169 |
Goemai | -0.124 | -0.138 | -0.11 | 0.063 | 0.052 | 0.074 |
Gorwaa | -0.125 | -0.136 | -0.114 | 0.009 | 0 | 0.018 |
Hoocąk | -0.099 | -0.109 | -0.09 | 0.141 | 0.132 | 0.151 |
Jahai | -0.062 | -0.073 | -0.05 | 0.142 | 0.131 | 0.153 |
Jejuan | -0.093 | -0.104 | -0.083 | 0.038 | 0.028 | 0.049 |
Kakabe | -0.123 | -0.135 | -0.111 | 0.103 | 0.091 | 0.115 |
Kamas | -0.096 | -0.111 | -0.082 | 0.003 | -0.012 | 0.017 |
Komnzo | -0.081 | -0.091 | -0.071 | 0.026 | 0.018 | 0.035 |
Light Warlpiri | -0.144 | -0.154 | -0.133 | 0.078 | 0.067 | 0.088 |
Lower Sorbian | -0.078 | -0.089 | -0.067 | 0.046 | 0.037 | 0.056 |
Mojeño Trinitario | -0.147 | -0.156 | -0.137 | -0.074 | -0.083 | -0.065 |
Movima | -0.014 | -0.023 | -0.006 | -0.053 | -0.062 | -0.045 |
Nafsan (South Efate) | -0.072 | -0.082 | -0.062 | 0.094 | 0.085 | 0.103 |
Nisvai | -0.064 | -0.074 | -0.054 | 0.151 | 0.144 | 0.159 |
Nllng | -0.098 | -0.111 | -0.086 | 0.155 | 0.142 | 0.167 |
Northern Alta | -0.069 | -0.079 | -0.059 | 0.091 | 0.081 | 0.101 |
Northern Kurdish (Kurmanji) | -0.022 | -0.033 | -0.011 | 0.033 | 0.023 | 0.042 |
Pnar | -0.021 | -0.035 | -0.007 | 0.087 | 0.076 | 0.099 |
Resígaro | -0.079 | -0.089 | -0.069 | 0.012 | 0.003 | 0.022 |
Ruuli | -0.039 | -0.048 | -0.03 | 0.069 | 0.061 | 0.078 |
Sadu | -0.124 | -0.136 | -0.112 | 0.2 | 0.189 | 0.212 |
Sanzhi Dargwa | -0.135 | -0.148 | -0.122 | 0.012 | 0 | 0.023 |
Savosavo | -0.052 | -0.062 | -0.042 | -0.125 | -0.134 | -0.117 |
Sümi | -0.09 | -0.101 | -0.08 | 0.109 | 0.098 | 0.12 |
Svan | -0.066 | -0.074 | -0.057 | -0.026 | -0.035 | -0.017 |
Tabaq (Karko) | -0.213 | -0.223 | -0.203 | -0.071 | -0.08 | -0.061 |
Tabasaran | -0.138 | -0.151 | -0.125 | -0.025 | -0.037 | -0.012 |
Teop | -0.17 | -0.181 | -0.159 | 0.072 | 0.063 | 0.081 |
Texistepec Popoluca | -0.07 | -0.081 | -0.059 | 0.02 | 0.011 | 0.03 |
Urum | -0.126 | -0.134 | -0.118 | -0.007 | -0.015 | 0.001 |
Vera'a | -0.152 | -0.163 | -0.141 | 0.169 | 0.16 | 0.177 |
Warlpiri | -0.147 | -0.157 | -0.137 | -0.026 | -0.035 | -0.017 |
Yali (Apahapsili) | -0.192 | -0.203 | -0.18 | 0.14 | 0.13 | 0.15 |
Yongning Na | -0.128 | -0.142 | -0.114 | 0.098 | 0.085 | 0.112 |
Yucatec Maya | -0.067 | -0.079 | -0.056 | -0.075 | -0.084 | -0.066 |
Yurakaré | -0.198 | -0.204 | -0.191 | -0.035 | -0.04 | -0.029 |
Figure 2: The 95% confidence intervals for the effect of sequence length (top) and position (bottom) on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of words within sentences.
James et al. (1) recently found that Menzerath’s law can be detected in pseudorandom sequences of birdsong syllables that are forced to match the durations of real songs. James et al. (1) interpret their model as approximating simple motor constraints, while stronger effects in the real data would indicate additional mechanisms (e.g., communicative efficiency through behavioral plasticity). I originally planned to compare the strength of Menzerath’s law in the real data with simulated data from the model of James et al. (1), as I recently did for house finch song (2), but analyses of language data suggest that it is far too conservative of a null model. 0 of the 51 of languages in the DoReCo dataset exhibit Menzerath’s law to a greater extent than simulated data. Even though many whale species exhibit Menzerath’s law to a greater extent than simulated data from the null model of James et al. (1) (75%; 12 out of 16 species), I do not want to over-interpret this result given the pattern in the human data. Upon further reflection I think that the fundamental assumption of James et al. (1), that sequence durations are governed by motor constraints alone, is unlikely to apply to many species with more complex communication systems. In humpback whales and sperm whales, for example, there appears to be significant inter-individual variation in song and coda length depending on social context (3,4). More details about this analysis are below.
The production constraint model of James et al. (1) works as follows. For each iteration of the model, a pseudorandom sequence was produced for each real song in the dataset. Syllables were randomly sampled (with replacement) from the population until the duration of the random sequence exceeded the duration of the real song. If the difference between the duration of the random sequence and the real song was <50% of the duration of the final syllable, then the final syllable was kept in the sequence. Otherwise, it was removed. Each iteration of the model produces a set of random sequences with approximately the same distribution of durations as the real data.
For each species, I generated 100 simulated datasets from the (1) random sequence model and the (2) production constraint model. Then, I fit Menzerath’s law separately to each of the 100 simulated datasets and pooled the model estimates for \(a\) and \(b\) using Rubin’s rule as implemented in the mice package in R (5). The results can be seen in Figure 3.
Most importantly, the estimated effects from the production constraint model tend to be more negative than those from the real human language data, suggesting that this null model is far too conservative to be informative about “language-like” efficiency.
Figure 3: The point estimates from the real data alongside 95% confidence intervals from 10 simulated datasets from the production constraint model, for the effect of sequence length on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of phonemes within words.
Figure 4: The point estimates from the original datasets (red) compared to median-interpolated datasets (blue). Interpolating sequences with the median duration of each element category appears to systematically shift model estimates towards zero (in over 90% of cases).